## Reasonable Solutions

I believe teaching multiple grade levels within the same day has value. Being able to observe how students think about numbers and the strategies that they use over time gives teachers a different perspective.  It also shows some of the linear progression of math skills and strategies. I found this especially evident as I read through Kathy Richardson’s book during July. I currently serve as a math teacher for students in grades 2-5.  I get to see how students progress over time and what tends to trip them up.  I also see the problems that emerge when students start to rely on tricks and formulas before having a deep understanding of a particular concept.  One thing that I also continue to observe is that students sometimes struggle to be reasonable with their estimates. Part of that may be due to an over-reliance on algorithms and the other part may relate to exposure. Students aren’t given (or take the) time to reflect and ask themselves whether the answer truly makes sense or not.  This tells me that students are relying on a prescribed process or algorithm and reasonableness comes second.

In an effort to move towards reasoning, I’ve been using Estimation 180 on a daily basis.  I feel that the class is become better at estimating and their justification has improved.  Making sense with number puzzles also seem to be helping students create reasonable estimates and solutions.  Basically, students are given a story that has blanks.

Students are then are given a number bank. Sometimes too many numbers are in the bank.

Then students have to justify why they picked each answer.  This can be completed in verbal or written form.

Usually I have students explain their reasoning with a partner.  The class has completed a number of these types of making sense with numbers puzzles.  I can say that students are now looking more closely at the magnitude of the actual numbers before estimating or finalizing an answer. That’s progress and I’m confident that students are more willing to use that strategy along their math journey in the future.

## Understanding Number Relationships

Today I was able to dig a bit deeper into Kathy Richardson’s book. The first chapter was related to counting and critical phases that are needed as students develop numeracy skills.  The second chapter focuses on number relationships. In order for students to compare numbers they need to be able to distinguish between larger and smaller. Once at this stage students can recognize that numbers are found within numbers.  For example, eight is found within 10. When comparing numbers students generally start to identify differences between the original number and one.

Richardson states that being able to change a number by counting on or adding to a group is a Critical Learning Phase.  Counting on or adding to a group of numbers is a strategy students use when comparing numbers. I believe primary students might use this strategy to find out how many blocks are in both stacks below.

Assume that each block is the same size.  Now, how do you think primary students would count these two different stacks?  The strategy that they may use to solve this can tell  more about their understanding.  Are they counting each block individually or using the first stack to count on to find the second stack? While comparing numbers younger students often count each block individually.  The model below shows a different strategy.

In this case the student has taken the five and built upon it to find four more.  The five and four are nine.  This student didn’t count each stack individually.

I find this interesting as it may apply to other areas of mathematics.  After reading this I started to think of how increasing the complexity could apply to fraction concepts.  Specifically, I thought of how theses blocks and fractions are similar:

If the green stack is one whole what is the second stack’s value?  How would your students solve this?  In the example students may identify that each block is 1/5.  When looking at the parts on the right they might start off at 5/5 and add to that particular block line. Fractions can lead to confusion with a non-linear scale being present.  This is especially the case if students are always seeing 1:1 ratio when counting objects.

I thought a number line might be a better representations for a fraction problem. Richardson notes that number lines are only symbolic relationships.  She also states that when students use number lines they’re most likely not thinking of quantities, but more so using the line to find the solution. They’re using it as a tool to count on to find a solution.  Number lines are used frequently at the early elementary levels so this is something I’m going to keep in mind for the new school year.